MARS Equation
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"Maximized Asymptotic Return System "

# The MARS Equation

#### A Guide to Succesful Money Managment By Douglas McCormack September 1986

Most gamblers and investors use their intuition to solve the tricky problem of money managment:
What portion of your bankroll (assets) should be risked on any one bet (investment)?
The purpose of this paper is to describe the mathematical equation that can be used to optimize you money management. The examples will show some practical uses of the equation, and will give you a chance to test your intuition against the equation.

#### Derivation of the MARS Equation

The equation is called the "MARS Equation", standing for "Maximized Asymptotic Return System". It is closely related to the Kelly betting system, where the gambler always bets a fixed percentage of his current bankroll. As the bankroll grows, his bets will become larger. If the bankroll shrinks, his bets will become smaller. The Kelly bettor will always bet a fraction of what he has left, so theoretically he will never go broke. The MARS equation is simply a way to calculate the optimal Kelly bet. Any gambling game becomes more predictable as one averages his results over an increasing number of plays. For instance, it would be very difficult to assess the fairness of a coin if one tried only ten flips. But after a thousand flips, the coins fairness could be stated with confidence. It would be quite easy to get seven heads out of ten flips, but it would be almost impossible to get more than 700 heads out of 1000 flips. After a million flips, the percentage of heads should be very close to 50%. After an infinite number of flips, the percentage of heads should be exactly equal to 50%. A mathematician would say the percentage of heads becomes "asymptotic" (infinitly close) to 50% as the number of flips is increased.

Suppose you know of a bet (investment) that guarantees you a 20% profit every time you play. Let us define a number R as your return rate = 1.2 in this case. This means that your bet is multiplied by 1.2 every time you play. If you start with \$100 and bet your entire bankroll on every play, after three plays you will have:

\$100 * 1.2 * 1.2 * 1.2 = \$100 * 1.2 3
Which comes to \$172.80. This is simply an example of compound interest.

Likewise, if you knew of a game that guaranteed a 20% loss every time you played, then R=0.8.
After three plays you will have:

\$100 * 0.8 * 0.8 * 0.8 = \$100 * 0.8 3
Which comes to \$51.20.

The following table clearly demonstrates the difference between these two games:

 R=1.20 R=0.80 After 1 play \$100 * (1.21) = \$120.00 \$100 * (0.81) = \$80.00 After 2 plays \$100 * (1.22) = \$144.00 \$100 * (0.82) = \$64.00 After 3 plays \$100 * (1.23) = \$172.80 \$100 * (0.83) = \$51.20 After 10 plays \$100 * (1.210) = \$619.17 \$100 * (0.810) = \$10.74 After 20 plays \$100 * (1.220) = \$3833.76 \$100 * (0.820) = \$1.15
Now let us consider a more realistic game. Suppose we flip a coin, and we get R of 1.2 for heads and R of 0.8 for tails. After a million flips, we can be quite certain that we have obtained ½ heads and ½ tails.
Overall return for a million flips:
[1.2 * 1.2 * 1.2 * ...(½ million times)] * [.8 * .8 * .8 * ...(½ million times)]
= [1.2500,000] * [.8500,000]
Notice that the actual order of the wins and losses has no has no effect on the overall return. The average return per flip is the one millionth root of the above overall return.
Average return per flip:
[1.2500,000 * .8500,000](1/1,000,000)
= [1.2.5] * [.8.5]
Which comes to 0.98.

Therefore, the long term player should expect his bankroll to shrink at an average rate or 2% per play. He should expect to lose money on this seemingly fair game. The longer he plays, the closer he will become to experiencing the asymptotic return rate of 0.98.

Now, let us generalize the above specific example:

Let W represent the return rate if you win.
Let p represent your probability of winning.
Let L represent the return rate if you lose.
Let q represent your probability of losing.
Let B represent return rate for any unbet money (bank interest)
Let X represent what portion of your money is bet on each play.
Let R represent your asymptotic return rate.

Assume p + q = 1 (no ties, you either win or lose)
Now, following the above example:
R=[{return on a win}p] * [{return on a loss}q]
R=[{(1-X)B + WX}p] * [{(1-X)B + LX}q]
(Equation #1)
This equation gives the asymptotic return rate "R" that results from using money management strategy "X" in playing a game under conditions {W,p,L,q,B}.
The obvious question is "what X will produce the maximum value of R ?"
A mathematician can answer such questions by using the technique of differential calculus. He will rearrange equation #1 to tell you:
Maximum R occurs when X = [L + (W-L)p - B] / [(W+L) - WL/B - B]
(Equation #2)
This is the MARS equation

# Pratical Uses of the MARS Equation

Example #1 ~ A Fair Game:
Imagine a casino that has coin flipping game; paying double or nothing; you call heads or tails.
For this game:
W=2     p=.5      B=1      L=0      q=.5
Using the MARS equation,
Optimal X = [0 + (2-0).5 - 1] / [(2+0) - (2*0)/1 - 1]
= 0
This proves the optimal way to play a fair game is not to bet. You should only bet when the game is in your favour.
Example #2 ~ A Favorable Game:
Imagine you study the action of the above game and you discover a way to correctly predict the outcome 60% of the time. Here is a favorable situation! But how should you play it? If you bet 100% of you money on every play, you will go broke on the first loss. If you bet only pennies, you will be wasting time.
For this game:
W=2     p=.6      B=1      L=0      q=.4
Using the MARS equation,
Optimal X = [0 + (2-0).6 - 1] / [(2+0) - (2*0)/1 - 1]
= .20
Thus you should bet 20% of your current bankroll on every play. Assuming you start with \$100 bankroll, how quickly will your money grow?
Using Equation #1:
R = [{(1-.2)1 + 2*.2}.6] * [{(1-.2)1 + 0*.2}.4]
= 1.0203
Thus you can expect your initial \$100 bankroll to grow at an average rate or 2.03% per play. If you stayed for 100 plays, always betting 20% of your current bankroll, your theoretical outcome would be \$100 * (1.023100) which comes to \$749. The longer you play, the more certain you can be of achieving this therotical outcome.
Example #3 ~ A Favorable Game:
Imagine example #1 is modified so the casino pays triple or nothing.
For this game:
W=3     p=.5      B=1      L=0      q=.5
Using the MARS equation,
Optimal X = [0 + (3-0).5 - 1] / [(3+0) - (3*0)/1 - 1]
= .25
Thus you should bet 25% of your current bankroll on every play. How quickly will your money grow?
Using Equation #1:
R = [{(1-.25)1 + 3*.25}.5] * [{(1-.25)1 + 0*.25}.5]
= 1.0607
Thus you can expect your money to grow at an average compound rate of 6.07% per play.
Example #4 ~ A Lottery:
Imagine you are invited to buy a lottery tickets and your chance of winning is one in a million. Tickets cost \$1 each. Because of previous unclaimed prizes, the winner of this lottery will be paid \$5 million. This is clearly an attractive situation; but what should you pay for a ticket?
For this game:
W=500,000     p=1/1,000,000      B=1      L=0      q=999,999/1,000,000
Using the MARS equation,
Optimal X = [0 + (5,000,000-0).000001 - 1] / [(5,000,000+0) - (5,000,000*0)/1 - 1]
= .0000008
Thus if your entire net worth is \$200,000, you should pay \$200,000 * .0000008 = 16¢ for a single ticket (See comment section below).
Example #5 ~ A Horse Race:
An experienced handicapper estimates that a horse has a 40% chance of winning a race. The horse has odds of 3 to 1 (one dollar bet will produce a \$3 profit).
For this game:
W=(odds +1) = 4     p=.4      B=1      L=0      q=.6
Using the MARS equation,
Optimal X = [0 + (4-0).4 - 1] / [(4+0) - (4*0)/1 - 1]
= .20
Thus the handicapper should wager 20% of his bankroll on this horse.

The MARS equation is so very useful at the racetrack, we will restate the equation in a special handicapper's form:

Wager in % of bankroll = {[(Odds +1) * chance of winning -1] / Odds} * 100%
Example #6 ~ The Stock Market:
Imagine you are considering buying stock of a small mining exploration company. If the company finds ore, the stock will be tripled a year from now. But if no ore is found, the stock will be at one third of its current value. The chance of finding ore is 50%. You can make 10% annual interest on your money by simply leaving it in the bank.
For this game:
W=3     p=.5      B=1.10      L=.33      q=.5
Using the MARS equation,
Optimal X = [.33 + (3-.33).5 - 1.10] / [(3+.33) - (3*.33)/1.10 - 1.10]
= .43
Thus you should invest 43% of your portfolio in the stock and leave 57% in the bank.

Now suppose the bank raises its interest rates to 15%. How should you react?

B increases from 1.10 to 1.15
Optimal X = [.33 + (3-.33).5 - 1.15] / [(3+.33) - (3*.33)/1.15 - 1.15]
= .39
Thus you should sell some of your stock, so that your bank account is increased from 57% of your portfolio to 61%.
Example #7 ~ A Business Investment:
Imagine you are thinking about starting a small business. What is the optimal size of bussiness to start? In a good year, the company will make a profit of 35% on the current investment; while in a bad year the company will lose 20% of the current investment. Six years out of ten will be good years and four years will be bad. You have no way of predicting whether a given year will be good or bad. You only know there is a 60% chance of having a good year. You can make 10% annual interest on your money by simply leaving money in the bank.
For this game:
W=1.35     p=.6      B=1.10      L=.80      q=.4
Using the MARS equation,
Optimal X = [.80 + (1.35-.80).6 - 1.10] / [(1.35+.80) - (1.35*.80)/1.10 - 1.10]
= .44
Thus you should invest 44% of your money into the business and leave a cash reserve of 56% in the bank collecting interest. This cash reserve will tide you over in case of a string of bad years.
Example #8 ~ A Business Investment:
Suppose example # 7 is modified so there is a 70% chance of a good year.
For this game:
W=1.35     p=.7      B=1.10      L=.80      q=.3
Using the MARS equation,
Optimal X = [.80 + (1.35-.80).7 - 1.10] / [(1.35+.80) - (1.35*.80)/1.10 - 1.10]
= 1.25
With X greater than 100%, MARS is telling you to invest all of your money, plus 25% additional money borrowed at 10% interest rate. Your business should have one part debt to four parts equity. Your optimal debt / equity ratio is .25.

# Comments on Using the MARS Equation

These examples have shown that the MARS equation can be used in a variety of situations to provide useful guidelines for money management. The equation is quite easy to use, and can be readily adapted to specific game situations. However, some comments are in order:
• If several simultaneous bets are available, do not attempt to use MARS to portion your money between these bets. The MARS general principles should apply, but the mathematics becomes explosively complex, even for two or three bets. The problem becomes one of portfolio optimization (see bibliolgrahy).
• When the optimal bet is less than zero, do not bet; but attempt to find some way to get onto the opposite side of this bet. For example, in the stock market, a negative X is telling you to sell short.
• The MARS equation is not valid in cases where the denominator is less than or equal to zero.
• If you bet lower than the MARS optimal amount, your asymptotic growth will remain positive and you money will grow at a smoother (less volitile) though slower rate.
• If you over bet the MARS optimal amount, your asymptotic growth could become negative and there will be a high volitility in the size of your bankroll. Even an favorable game can result in negative growth if you insist on heavily over betting the MARS optimum. When in doubt, you should under bet the MARS optimum.
• Sometimes your participation in a game affects your probabilities or payback. If you bet huge amounts at a horse race, the odds will be changed by your bet. If you purchase many lottery tickets, your odds of winning increase. Refering to example #4 above, the best strategy would be to purchase all of the million tickets which guarantees a \$4 million profit. In these cases, the values for L, W, p, q may be adjusted to reflect their changed values.
• In most betting games L is zero; while in most investments L is greater than zero. Whenever L is zero, the optimal bet size "X" is always less than the probability of winning "p", regardless of the winning payback "W". This fact provides a useful rule-of-thumb:
"Never bet a portion of your bankroll that exceeds you chance of winning"

# Conclusion

Mathematics can be fun when it is used to solve practical, everyday problems. Try adapting the MARS equation to your favorite game. Explore the effects of probabilities, paybacks and interest rates. You will gain new understanding and feeling for the sensitivities of your game. Perhaps new innovative strategies will emerge. Have fun!

#### Bibliography

• Henry A. Latane. "Investment Criteria - A Three Asset Portfolio Balance Model." The Review of Economics and Statistics, Vol. XLV, No. 4 (November, 1963), pp. 427-430.
• J. L. Kelly, Jr.. "A New Interpretation Of The Information Rate." The Bell System Technical Journal, July 1956, pp. 917-925.
• Harry M. Markowitz. "Portfolio Selection - Efficient Diversification Of Investments." Cowles Foundation, Yale University, 1959 / 1970.

#### Acknowledgement

The author wishes to acknowledge the assistance of Mr. Rodger Hui in performing the necessary calculus to get from equation #1 to equation #2.